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We construct the four-variable primitive $p$-adic $L$-functions associatedwith the triple product of Hida families and prove the explicit interpolationformulae at all critical values in the balanced range. Our construction is tocarry out the $p$-adic interpolation of Garrett's integral representation oftriple product $L$-functions via the $p$-adic Rankin-Selberg convolutionmethod. As an application, we obtain the cyclotomic $p$-adic $L$-function forthe motive associated with the triple product of elliptic curves and prove thetrivial zero conjecture for this motive.

Four-variable $p$-adic triple product $L$-functions and the trivial zero conjecture